\contentsline {chapter}{List of Tables}{iv}
\contentsline {chapter}{List of Figures}{v}
\contentsline {chapter}{\numberline {1}Introduction}{1}
\contentsline {section}{\numberline {1.1}Contributions}{2}
\contentsline {section}{\numberline {1.2}Proposal Structure}{2}
\contentsline {section}{\numberline {1.3}Related Work}{3}
\contentsline {chapter}{\numberline {2}Two-Stage Capacity Expansion in Survivable Networks}{4}
\contentsline {section}{\numberline {2.1}Problem Statement}{4}
\contentsline {subsection}{\numberline {2.1.1}Components of the problem}{6}
\contentsline {subsection}{\numberline {2.1.2}Statement of the problem}{6}
\contentsline {section}{\numberline {2.2}Modeling Approach}{7}
\contentsline {subsection}{\numberline {2.2.1}Modeling Assumptions}{7}
\contentsline {paragraph}{Introductory Example}{8}
\contentsline {subsection}{\numberline {2.2.2}Modeling Assumptions}{9}
\contentsline {paragraph}{Assumption 1}{11}
\contentsline {paragraph}{Assumption 2}{11}
\contentsline {paragraph}{Assumption 3}{11}
\contentsline {section}{\numberline {2.3}Model Formulation}{11}
\contentsline {subsection}{\numberline {2.3.1}Illustration of the model formulation for the example network}{13}
\contentsline {subsection}{\numberline {2.3.2}Scaling and Computational Complications}{15}
\contentsline {chapter}{\numberline {3}Solution Approach To Capacity Expansion Problem in Survivable Networks}{17}
\contentsline {section}{\numberline {3.1}Feasibility Test}{17}
\contentsline {section}{\numberline {3.2}Finding All Possible Paths between Two Nodes}{17}
\contentsline {section}{\numberline {3.3}Reduction of Solution Space}{17}
\contentsline {paragraph}{Proposition 1}{18}
\contentsline {paragraph}{\indent }{18}
\contentsline {paragraph}{Example 1}{20}
\contentsline {paragraph}{Proposition 2}{21}
\contentsline {paragraph}{Example 2}{22}
\contentsline {section}{\numberline {3.4}Bounds on the Objective Function }{24}
\contentsline {subsection}{\numberline {3.4.1}Upper Bounds on the Cost Function}{24}
\contentsline {paragraph}{Lemma 1}{24}
\contentsline {paragraph}{Proposition 3}{24}
\contentsline {paragraph}{Example 3}{25}
\contentsline {paragraph}{Proposition 4}{26}
\contentsline {paragraph}{\indent }{26}
\contentsline {paragraph}{Example 4}{27}
\contentsline {subsection}{\numberline {3.4.2}Lower Bounds on the Cost Function}{29}
\contentsline {paragraph}{Proposition 5}{29}
\contentsline {paragraph}{\indent }{29}
\contentsline {paragraph}{Example 5}{30}
\contentsline {paragraph}{Lemma 2}{30}
\contentsline {paragraph}{\indent }{30}
\contentsline {paragraph}{Example 6}{30}
\contentsline {section}{\numberline {3.5}Decomposition Approach}{31}
\contentsline {subsection}{\numberline {3.5.1}The Master Problem}{31}
\contentsline {subsection}{\numberline {3.5.2}The Sub-Problems}{32}
\contentsline {subsection}{\numberline {3.5.3}Constructing Feasibility Cuts}{33}
\contentsline {subsection}{\numberline {3.5.4}Decomposition Algorithm}{34}
\contentsline {chapter}{\numberline {4}Experimental Results}{36}
\contentsline {section}{\numberline {4.1}Extensions to the Test Instances}{36}
\contentsline {paragraph}{Failure Scenarios}{36}
\contentsline {paragraph}{Capacity Expansion Decisions}{37}
\contentsline {section}{\numberline {4.2}Experimental Settings}{37}
\contentsline {section}{\numberline {4.3}Evaluating the Performance of Decomposition Algorithm}{37}
\contentsline {subsection}{\numberline {4.3.1}Experiments with Single Failure Scenarios - Single Fault}{37}
\contentsline {paragraph}{Discussion}{39}
\contentsline {subsection}{\numberline {4.3.2}Experiments with Single Failure Scenarios - Multiple Faults}{40}
\contentsline {paragraph}{Discussion}{40}
\contentsline {chapter}{\numberline {5}Case Study: \\ Survivable Capacity Expansion in Iraninan Oil Products Distribution Network}{44}
\contentsline {section}{\numberline {5.1}Background}{44}
\contentsline {section}{\numberline {5.2}Distribution Model - Over View}{45}
\contentsline {subsection}{\numberline {5.2.1}Products}{45}
\contentsline {subsection}{\numberline {5.2.2}Refineries as supply nodes}{46}
\contentsline {subsection}{\numberline {5.2.3}Means of Transportation}{48}
\contentsline {section}{\numberline {5.3}Other Products Transportation Activities}{48}
\contentsline {subsection}{\numberline {5.3.1}Bunkering}{49}
\contentsline {subsection}{\numberline {5.3.2}Swap}{49}
\contentsline {section}{\numberline {5.4}Modeling the problem}{50}
\contentsline {subsection}{\numberline {5.4.1}Nodes and Links}{50}
\contentsline {subsection}{\numberline {5.4.2}Supply and Demand}{50}
\contentsline {subsection}{\numberline {5.4.3}Failure Scenarios}{50}
\contentsline {subsection}{\numberline {5.4.4}Data Entities}{50}
\contentsline {subsection}{\numberline {5.4.5}Supply Data : Production and Import }{50}
\contentsline {subsection}{\numberline {5.4.6}Bunkering and Swap Data}{50}
\contentsline {subsection}{\numberline {5.4.7}Demand Data : Consumption Data by Product}{50}
\contentsline {subsection}{\numberline {5.4.8}Alternatives: Costs and Capacities}{50}
\contentsline {chapter}{\numberline {6}Inexact Solution Methods To Capacity Expansion Problem in Survivable Networks }{51}
\contentsline {section}{\numberline {6.1}Feasibility Check}{51}
\contentsline {subsection}{\numberline {6.1.1}Feasibility Test}{52}
\contentsline {paragraph}{Lemma 3}{52}
\contentsline {paragraph}{Example 1}{52}
\contentsline {section}{\numberline {6.2}Optimality Gap}{52}
\contentsline {section}{\numberline {6.3}Start Heuristics}{54}
\contentsline {subsection}{\numberline {6.3.1}Costliest Feasible Solution}{54}
\contentsline {paragraph}{Example 2}{54}
\contentsline {subsection}{\numberline {6.3.2}Worst Case Scenario}{56}
\contentsline {paragraph}{Example 3}{56}
\contentsline {subsection}{\numberline {6.3.3}Rounding Heuristic}{56}
\contentsline {subsubsection}{Capacity Based Rounding Algorithm-CBRA}{57}
\contentsline {paragraph}{\indent \textit \textbf {{Proof :}}}{58}
\contentsline {paragraph}{Example 4}{59}
\contentsline {subsubsection}{Capacity Based Rounding Algorithm-Modified (CBRAM)}{59}
\contentsline {subsection}{\numberline {6.3.4}Evaluating Start Heuristics}{60}
\contentsline {section}{\numberline {6.4}Improvement Heuristics}{60}
\contentsline {subsection}{\numberline {6.4.1}Neighborhood Search Algorithms - Moving From Feasibility to Optimality}{61}
\contentsline {subsection}{\numberline {6.4.2}NS-1 Algorithm}{61}
\contentsline {subsection}{\numberline {6.4.3}NS-2 Algorithm}{62}
\contentsline {subsection}{\numberline {6.4.4}NS-3 Algorithm}{62}
\contentsline {subsection}{\numberline {6.4.5}Neighborhood Search Algorithms - Moving From Optimality to Feasibility}{63}
\contentsline {subsubsection}{NS-4 Algorithm}{63}
\contentsline {subsection}{\numberline {6.4.6}Comparing Heuristic Methods}{64}
\contentsline {chapter}{References}{66}
\contentsline {chapter}{\numberline {7}Lorem Ipsum}{67}
\contentsline {section}{\numberline {7.1}A First Section}{67}
